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Explore the concept of Spatial Systems and Cellular Automata (CAs), their structure, rules, and applications in various fields such as computer architecture, physical phenomena, and quantum computation. Learn about the famous Conway's Game of Life and its fascinating behavior. Investigate the emergence of complex dynamics in CAs under different conditions.
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Cellular Automata (CAs) • Invented by von Neumann in 1940s to study reproduction • He succeeded in constructing a self-reproducing CA • Have been used as: • massively parallel computer architecture • model of physical phenomena (Fredkin, Wolfram) • Currently being investigated as model of quantum computation (QCAs)
Structure • Discrete space (lattice) of regular cells • 1D, 2D, 3D, … • rectangular, hexagonal, … • At each unit of time a cell changes state in response to: • its own previous state • states of neighbors (within some “radius”) • All cells obey same state update rule • an FSA • Synchronous updating
Example:Conway’s Game of Life • Invented by Conway in late 1960s • A simple CA capable of universal computation • Structure: • 2D space • rectangular lattice of cells • binary states (alive/dead) • neighborhood of 8 surrounding cells (& self) • simple population-oriented rule
State Transition Rule • Live cell has 2 or 3 live neighbors stays as is (stasis) • Live cell has < 2 live neighbors dies (loneliness) • Live cell has > 3 live neighbors dies (overcrowding) • Empty cell has 3 live neighbors comes to life (reproduction)
Demonstration of Life Run NetLogo Life or <www.cs.utk.edu/~mclennan/Classes/420/NetLogo/Life.html> Go to CBNOnline Experimentation Center <mitpress.mit.edu/books/FLAOH/cbnhtml/java.html>
Some Observations About Life • Long, chaotic-looking initial transient • unless initial density too low or high • Intermediate phase • isolated islands of complex behavior • matrix of static structures & “blinkers” • gliders creating long-range interactions • Cyclic attractor • typically short period
From Life to CAs in General • What gives Life this very rich behavior? • Is there some simple, general way of characterizing CAs with rich behavior? • It belongs to Wolfram’s Class IV
Wolfram’s Classification • Class I: evolve to fixed, homogeneous state ~ limit point • Class II: evolve to simple separated periodic structures ~ limit cycle • Class III: yield chaotic aperiodic patterns ~ strange attractor (chaotic behavior) • Class IV: complex patterns of localized structure ~ long transients, no analog in dynamical systems
Langton’s Investigation Under what conditions can we expect a complex dynamics of information to emerge spontaneously and come to dominate the behavior of a CA?
Approach • Investigate 1D CAs with: • random transition rules • starting in random initial states • Systematically vary a simple parameter characterizing the rule • Evaluate qualitative behavior (Wolfram class)
Why a Random Initial State? • How can we characterize typical behavior of CA? • Special initial conditions may lead to special (atypical) behavior • Random initial condition effectively runs CA in parallel on a sample of initial states • Addresses emergence of order from randomness
Assumptions • Periodic boundary conditions • no special place • Strong quiescence: • if all the states in the neighborhood are the same, then the new state will be the same • persistence of uniformity • Spatial isotropy: • all rotations of neighborhood state result in same new state • no special direction • Totalistic [not used by Langton]: • depend only on sum of states in neighborhood • implies spatial isotropy
Langton’s Lambda • Designate one state to be quiescent state • Let K = number of states • Let N = 2r + 1 = size of neighborhood • Let T = KN = number of entries in table • Let nq = number mapping to quiescent state • Then
Range of Lambda Parameter • If all configurations map to quiescent state:l = 0 • If no configurations map to quiescent state:l = 1 • If every state is represented equally:l = 1 – 1/K • A sort of measure of “excitability”
Example • States: K = 5 • Radius: r = 1 • Initial state: random • Transition function: random (given l)
Demonstration of1D Totalistic CA Run NetLogo 1D CA General Totalistic or <www.cs.utk.edu/~mclennan/Classes/420/NetLogo/CA-1D-General-Totalistic.html> Go to CBNOnline Experimentation Center <mitpress.mit.edu/books/FLAOH/cbnhtml/java.html>
Class I (l = 0.2) time
Class IV Shows Some of the Characteristics of Computation • Persistent, but not perpetual storage • Terminating cyclic activity • Global transfer of control/information
l of Life • For Life, l ≈ 0.273 • which is near the critical region for CAs with: K = 2 N = 9
Shannon Information(very briefly!) • Information varies directly with surprise • Information varies inversely with probability • Information is additive • The information content of a message is proportional to the negative log of its probability
Entropy • Suppose have source S of symbols from ensemble {s1, s2, …, sN} • Average information per symbol: • This is the entropy of the source:
Maximum and Minimum Entropy • Maximum entropy is achieved when all signals are equally likely No ability to guess; maximum surprise Hmax = lg N • Minimum entropy occurs when one symbol is certain and the others are impossible No uncertainty; no surprise Hmin = 0
Entropy of Transition Rules • Among other things, a way to measure the uniformity of a distribution • Distinction of quiescent state is arbitrary • Let nk = number mapping into state k • Then pk = nk / T
Entropy Range • Maximum entropy (l = 1 – 1/K): uniform as possible all nk = T/K Hmax = lg K • Minimum entropy (l = 0 or l = 1): non-uniform as possible one ns = T all other nr = 0 (r s) Hmin = 0
Further Investigations by Langton • 2-D CAs • K = 8 • N = 5 • 64 64 lattice • periodic boundary conditions
